Friday, 6 September 2013

Postulates of Quantum Mechanics: State Evolution

This postulate describes the way quantum states can be transformed to other states. Given an isolated* state \(\left|\psi\right>\in\mathcal{H}\), the permissible transformations are given by a unitary operator \(U\):
 \[U: \mathcal{H} \rightarrow \mathcal{H},\]
 which maps \[\left|\psi\right>\mapsto U\left|\psi\right>.\]
Unitarity implies that \(UU^{\dagger}=I\), where \(U^{\dagger}\)  denotes the conjugate-transpose of \(U\), and \(I\) is the identity transformation. If \(\mathcal{H}\) is of dimension \(N\) with a suitable basis,  then \(U\) can be represented as an \(N\times N\) matrix. Unitary operators have inverses by definition which are given by the conjugate-transpose \(U^\dagger\), which is just the matrix that results by interchanging the rows of the matrix \(U\) with  its columns and taking the complex conjugate of each of its matrix coefficients. This unitary property is necessary to ensure that states of unit norm get mapped to states of unit norm.

*It is important to mention the use of the word "isolated" in the statement of the postulate given above. What is meant by an isolated system here is synonymous to a closed system. This is in contrast to an open system, which is one that interacts with a larger subsystem called the environment. In this way, the quantum system under consideration is a subsystem of the total system defined by that system and its external environment. This environment itself can be an arbitrary system that contains the main system under consideration. In the limiting case, the environment is essentially the universe at large (or in better interpretations that which is the multiverse).  A closed or isolated system is one that does not interact at all with any other containing system, and it is only in this special and highly idealized case that the unitary evolution stated in the postulate actually holds. In general and in practice, systems that are dealt with are actually open. In fact, there really is no such thing as a perfectly isolated system unless, of course, that system is the entire universe (or multiverse). For the purpose of analysis, as it is usually done in most general physical analysis, we will only consider the case of isolated systems here and their unitary dynamics.

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